1. Field of the Invention
This invention relates to optical gratings.
2. Related Art
An optical grating can be considered to be a sequence of grating lines. The lines modify the reflection and transmission characteristics of an optical transmission medium to which the grating is applied so allowing the characteristics to be tailored, to a greater or lesser degree, to a desired application. For example, an optical grating is used in a distributed feedback laser (DFB) to control the wavelength at which the laser is able to lase. In another application, an optical grating is used to control the transmission characteristics of an optical waveguide, for example an optical fiber.
An article titled "D-Fiber Grating Reflection Filters", P Yennadhiou and S. A. Cassidy, OFC 90 (1990) describes a D-fiber mounted on a flat substrate to expose the optical field in the fiber core. A holographically formed grating was placed on top of the substrate to give a periodic sequence of changes to the effective refractive index seen by the electric field. The changes in refractive index caused by the grating are very small but at each change in index there is a small amount of light reflected back down the fiber. At a certain resonant wavelength these small reflections build up through constructive interference to provide a large reflection whose magnitude is determined by the length of the grating and the size of the refractive index change. For a periodic grating with an arbitrary index profile this resonance occurs where the grating period is an integer multiple of half the wavelength, .lambda./2, divided by the mean effective index n.sub.0. In the special case when the index profile is a sequence of discrete jumps, the resonance only arises when the period is a odd multiple of .lambda./(2n.sub.0).
At wavelengths around the exact resonance, the reflection has a characteristics "sin (.lambda.)/.lambda." wavelength response profile of a finite-sized grating. The width of response peak is roughly inversely proportional to the grating length unless the reflectivity is very high. (see FIGS. 1(a) and 1(b)). When the peak reflectivity is high then multiple reflections become important and the reflection profile no longer narrows with increasing grating length. Instead the response flattens at around 100% reflectivity near the peak with very strong side lobes in the vicinity of the peak (see FIGS. 2(a) and 2(b)).
This characteristic profile is very difficult to change with conventional design methods. In particular, if the periodic change in effective refractive index is fixed by the material properties, then it is not possible to adjust the width of the wavelength response independently of the peak reflection. Nor is it possible by explicit design to remove the side lobe structure of smaller resonances on either side of the peak (although minor errors in the exact periodicity in the grating will often wash these out in practice).
Requirements have emerged which need reflection profiles that differ qualitatively from known prior art gratings. The first is to obtain a reflection profile that is flat over a comparatively large wavelength range (greater than about 1 nm wide) but with no side lobe reflections in the immediate neighborhood of this range. The peak reflection in this case is not important but it needs to be at least 10%. Such an optical grating could be positioned with an optical fiber network so that the connection with a central control could be checked by monitoring the reflections from an interrogation signal sent from the control center. The wavelength of the peak reflection would then be used to label the position of the grating and hence the integrity of the network could be checked at several places. A wide reflection is needed because the wavelength of the interrogation laser could not be accurately specified unless very expensive components were used. The side lobes need to be suppressed to prevent interference between different gratings in the network.
The second requirement is for a high reflection (as close to 100% as possible) in a narrow wavelength region, around 0.1 nm wide, with very low side lobes. This is for use as a wavelength selective mirror for use with a fiber laser to force it to operate in a narrow wavelength region only.
Other applications have been identified for nonconventional gratings where the wavelength response of the transmission and reflection properties could be specified. In particular, distributed Bragg reflectors (DBR) and distributed feedback lasers (DFB) appear to be very good candidates for such gratings.
It is a fairly straightforward matter, in principle, to calculate the effect on light travelling in one dimension of a sequence of steps in the effective index seen by this light. In a weakly guiding fiber waveguide both the electric field E and the magnetic field B are perpendicular to the direction of travel. The reflection and transmission coefficients are determined completely by the relation of E and B after passing through the region of index steps to their values before the region.
If the light passes a distance .DELTA.z through a region with a constant effective refractive index .beta., then ##EQU1## where .kappa. is the effective wavenumber, 2.pi..beta./.lambda., and ##EQU2## denotes the values of the electric and magnetic fields after a distance .DELTA.z. Hence if the light passes a distance .DELTA.z through a region of effective index .beta..sub.1, followed by a distance .DELTA.z.sub.2 through a region of effective index .beta..sub.2 then E and B are given by ##EQU3##
The effect of a sequence of small steps through the regions of differing refractive index can therefore be calculated from a scattering matrix, given by the product of all the small step matrices. Note that the matrix coefficients depend on the wavelength .lambda.. If the final scattering matrix S is given by ##EQU4## then the reflection coefficient is given by .vertline.R.vertline..sup.2 and the transmission coefficient by .vertline.T.vertline..sup.2 where ##EQU5## n.sub.o is the refractive index of the substrate and i=(-1).sup.1/2.
A 5 mm long grating with a pitch of say 0.25 .mu.m would have 20,000 steps and therefore the calculation for the scattering matrix would involve 20,000 matrix products. If the matrix were to be calculated at say 100 wavelengths in order to resolve the wavelength response of the grating, then the full scattering matrix of the grating would take several million arithmetic operations to calculate. This is therefore not a trivial calculation but one which would pose no difficulty for a reasonably powerful computer.
While the effect of a given sequence of steps in the effective index of the waveguide can easily be calculated, the converse task of designing the sequence to give the required properties to R and T is a different matter entirely. The problem lies in the number of calculations that have to be made. A crude approach of simply enumerating all the different possibilities, and testing each for its suitability, is out of the question: even if the grating pitch was constant and the changes were restricted to allowing a refractive index step or not, then the total number of possibilities would be around 220,000 so no amount of computer power would help.
In order to make any kind of optimization, the grating has to be defined in terms of some tractable number of parameters and repeated calculations made of how the grating properties change with these parameters. This basic idea is known. Simple parameters that have been used are grating pitch which may, for example, vary slowly along the grating's length to form a chirped grating or gratings that miss out some of the steps in a regular or smoothly varying fashion. See for example T Schrans, M Mittelstein and A Yariv "Tunable Active Chirped-Corrugation Waveguide Filters" Applied Physics Letters 55, 212-214 (1989) and D. C. J. Reid and C. M. Ragdale, I. Bennion, D. J. Robins, J. Buus and W. J. Steward "Phase-Shifted Moire Grating Fiber Resonators" Electronic Letters 36, 10-12 (1990), respectively.
These known types of optical gratings are not amenable to approaches of computation that give enough degrees of freedom of device characterization necessary to achieve the sort of wavelength response that are needed for many applications, for example as identified earlier in this application.